If you look it up, the very early days of the Korean war US planes used missiles but once they were expended, they became easy picking for the Russian pilots.
After learning the lesson the hard way, US planes switched back to guns.
I’m guessing that the early missiles were actually rockets.
Here’s some info from an unreliable source:
The Hughes AIM-4 Falcon was the first operational guided air-to-air missile of the United States Air Force.
Development began in 1946; the weapon was first tested in 1949.
The missile entered service with the USAF in 1956 (after the Korean war ended).
Some AiM-4s could have been fired in the Korean war on a test basis.
The Aim-4 did not have a proximity fuse. It was hit-to-kill.